There are many chemicals used in water treatment. It helps to think of water treatment as on big chemical reaction where each step is dependent on the performance of the preceding step. We add chemicals to drinking water to make it safe to drink. At a water treatment plant the chemicals are all of drinking water quality NSF/ANSI 60 and are not the same as what you would buy at a hardware store (unless you know exactly where to shop). Any chemical less than NSF 60 may and probably does contain other chemicals. Best example of this is sodium hypochlorite or bleach. Household bleach however also contains arsenic, not such a good thing to add to water you intend on drinking.
The following are broad categories of common water treatment chemicals and many chemicals in each group behave similarly.
Coagulants: The are chemicals used to assist the settling and filtration treatment steps. They destabilize the particles in the water and draw them into each other forming larger particles which then settle out or are filtered out. Coagulants are commonly based on aluminium, Aluminium Sulfate and Poly-aluminium Chloride being the two most popular ones. The health cost vs benefit of coagulants are they help remove every physical particle in the water including pathogenic organisms and the leave an aluminium residual. There are no known health risks associated with aluminium in drinking water.
Coagulant Aids: Are chemicals that assist the coagulants (shocking considering their name). Sometimes called polymers as they are essentially plastic. They form long stringy strands in the water and help collect particles formed around the coagulants and make the particles even bigger. Because they are plastic, there is no residual left in the water, they do not break down at all. I am almost tempted to say that 100% of the polymers get filtered out of the water, but I am not one to commit to absolute statements. Worst case scenario and you do manage to drink some of it, it isn’t toxic or absorbed by the body.
Disinfectants: I am planning a more detailed article on disinfection of water at a later date, so I will keep this one very general. Disinfection chemicals are inherently toxic. They are designed to kill living microscopic organisms. The trick with disinfectants is to use the minimum required to get the job done. Here the trade off is between the killing of pathogens which will cause severe disease/infection and a remote possibility of a long term toxic reaction. The other thing to consider while disinfecting is the formation of disinfection byproducts. Disinfection byproducts are suspected carcinogens. The byproducts are formed when the disinfectant contacts organic matter. Settling and filtering before disinfecting minimizes the formation of byproducts
Fluoride: This one gets an honorable mention as fluoride is absolutely necessary for healthy bones and teeth. Not enough The downside is, after 20 years of adding fluoride to public water sources, the results are in. The results suggest that fluoride in water has little affect on tooth health and bone density (which is why they were added in the first place). A lot of systems around north america have removed fluoride addition. I have not heard of a new facility built in the last ten years that has added fluoride (they may exist, I am just not aware of any and I have looked).
Softeners: Softeners are used to remove hard minerals from water and replace them with sodium or potassium. There are some long term negative health effects of too much sodium like hypertension. Too much potassium can lead to hyperkalemia, which can be fatal.
In general, the chemicals added are done so to prevent disease. Which usually outweighs the risk added from the chemicals themselves. Now you are a better informed to make that decision for yourself.